14 LOCALLY CONVEX SPACES 67 (12.14.11) Let E, E' be two locally convex spaces. Let Y be a directed set of seminorms defining the topology 0/E, and Tf a set of seminorms defining the topology ofE'. Let u : E -> E' be a linear mapping. Then the following conditions are equivalent: (a) u is continuous; (b) for each seminorm p' e F', there exists a neighborhood of 0 in E on which the function x\-+p'(u(x)) is bounded (or, equivalently (12.14.2), the semi- norm p' o u on E is continuous)', (c) for each seminorm p' e P, there exists a seminorm p e F and a real number c> 0 such thatp'(u(x)) <£ c -p(x)for allxeE. From the definition of the neighborhoods of 0 in E and E', condition (c) implies that u is continuous at the point 0, hence everywhere (12.8.4). So (c) implies (a). The condition (a) implies that the function p' ° u is continuous on E (because p' is continuous on E'), hence (b). Finally, suppose that (b) is satisfied. Then by hypothesis there exist two numbers r > 0 and a > 0 and a seminorm p e F such that the relation p(z) <j r implies p'(u(z)) <| a. Let x be any point of E and let A be a scalar strictly greater than 0 such that p(Xx) = Ap(x) g r. Then p'(u(kx)) g a, so that p'(u(x)) <; #/! If /?(jc) = 0, we can take 1 to be arbitrarily large, and therefore p'(u(x)) = 0; if XX) > 0> we ta^e A = r/p(x)9 and then we have p'(u(x)) <i (0/r) XX)- So in either case the condition (c) is satisfied, with c = a/r. Q.E.D. (12.14.11.1) Likewise, if u is a multilinear mapping of a product E! x E2 x - x Er of locally convex spaces into a locally convex space E', the continuity of u implies that for each seminorm p' e F' there exist neighborhoods Vy of 0 in E/ (1 ^y ^ «) such that the function (xl9 ..., #B)'-»Jp'(«(*i, - - , *)) is bounded in V{ x x Vn. If Fy is a directed set of seminorms defining the topology of Ey (1 £j£ri)9 this latter condition implies that there exist numbers a > 0, r7 > 0 (1 ^j ^n) and seminorms /^ 6 Fy (1 ^y :g /z) such that the relations pj(zj)£rj(l£J£ri) imply p'(u(zl9 ...,%))£ a. Now let (#!,..., ^) be any point of E! x x En , and for each j let Xj > 0 be such that /?y(Ay Xj) = Ajpfaj) ^ ry; then it follows that We deduce as above that ifpj(xj) = 0 for some index y, then lj may be chosen arbitrarily large, and hence p (u(xly . . . , xj) = 0. If on the other hand Pj(xj) ^ 0 for all y, we may take Aj = rj/pfaj) for 1 gy g «; putting c = a/rA . . . rB,ological vector spaces.